1. Equations of Trigonometric Functions

From what I know of trigonometric functions, they are repetitive ones. They just go on and on both sides of the x axis repeating the same shape after fixed intervals. Hence sine and cosine for example, have infinite solutions. Can anyone prove that (simple and easy as I am no pro or even something close) without a graph.
Another thing which intrigues me is that do these functions have an algebraic expression for them, behind the sine or cosine or tangent button on the calculator? Do they have any reality besides being a computer program? I mean that I can just plot a u-shaped parabola and program it into a computer as a function called, say, pine. On the Cartesian graph it is called y=pin (x). Whatever number you type for x will give a certain value of y until you end up with the aforementioned parabola, which really is just a graph of y=x^2 .Someone might ask so how do you compute y from x. Meaning the operations done on a value of x to obtain y, like powers, addition, multiplication, any constants...... This is what I am asking. For sin (x) do you have an expression with such operations whereby I can obtain the correct value of why, without using a calculator?

2. Re: Equations of Trigonometric Functions

The proof is pretty simple assuming you know some trig identities. In particular sin(A+B) = sin(A)cos(B)+sin(B)cos(A). Suppose we want x such that sin(x) = y. My claim is that sin(x+k*2pi) is also a solution for any k = 0,1,2. . .

Then sin(x+k2pi) = sin(x)cos(k2pi)+sin(k2pi)cos(x) = sin(x) (you should prove to yourself, using the aforementioned identity what sin and cosine of k2pi behave as they do). Thus you have an infinite number of solutions to your equation.

The second part of your question is not quite clear.

3. Re: Equations of Trigonometric Functions

Originally Posted by reindeer7
From what I know of trigonometric functions, they are repetitive ones.
You mean they are "periodic".

They just go on and on both sides of the x axis repeating the same shape after fixed intervals. Hence sine and cosine for example, have infinite solutions.
You mean they have an "infinite number of solutions". Some might interpret "they have infinite solutions" as meaning they have "infinity" as a solution.

Can anyone prove that (simple and easy as I am no pro or even something close) without a graph.
How you would prove that depends upon how they are defined. A standard (Calculus class) definition of sine and cosine is: "From the point (1, 0) on the unit circle (center at (0, 0), radius 1), measure around the circumference, counter clockwise, a distance "t" (if t is negative, go clockwise). "cos(t)" is defined as the x coordinate of the end point, "sin(t)" as the y coordinate. The fact that sine and cosine are periodic, with period $\displaystyle 2\pi$ follows from the fact that the circumference of a circle of radius 1 is $\displaystyle 2\pi$.

Another thing which intrigues me is that do these functions have an algebraic expression for them, behind the sine or cosine or tangent button on the calculator? Do they have any reality besides being a computer program? I mean that I can just plot a u-shaped parabola and program it into a computer as a function called, say, pine. On the Cartesian graph it is called y=pin (x). Whatever number you type for x will give a certain value of y until you end up with the aforementioned parabola, which really is just a graph of y=x^2 .Someone might ask so how do you compute y from x. Meaning the operations done on a value of x to obtain y, like powers, addition, multiplication, any constants...... This is what I am asking. For sin (x) do you have an expression with such operations whereby I can obtain the correct value of why, without using a calculator?
No, the trig functions cannot be calculated using a finite number of algebraic functions. They is why they are defined separately.